Discrete Calculus Applied Analysis on Graphs for Computational Scien
The field of discrete calculus, also known as "discrete exterior calculus", focuses on finding a proper set of definitions and differential operators that make it possible to operate the machinery of multivariate calculus on a finite, discrete space. In c
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Leo J. Grady Jonathan R. Polimeni
Discrete Calculus Applied Analysis on Graphs for Computational Science
Dr. Leo J. Grady Siemens Corporate Research 755 College Road East Princeton, NJ 08540-6632 USA [email protected]
Dr. Jonathan R. Polimeni Athinoula A. Martinos Center for Biomedical Imaging Department of Radiology Massachusetts General Hospital Harvard Medical School Charlestown, MA 02129 USA [email protected]
ISBN 978-1-84996-289-6 e-ISBN 978-1-84996-290-2 DOI 10.1007/978-1-84996-290-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010930985 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: KünkelLopka GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Leo (John) Grady dedicates the book to his very patient wife, Amy Grady, daughter Alexandra Grady, son Leo James Grady (who arrived during the course of writing) and the memory of his late father, Leo Joseph Grady.
Preface
The goal of this book is to present the topic of discrete calculus to scientists and engineers and to show how the theory can be applied to solving a wide variety of realworld problems. We feel that discrete calculus allows us to unify many approaches to data analysis and content extraction while being accessible enough to be widely applied in many fields and disciplines. This project initially began as a tutorial on discrete calculus and its applications, and we hope that this work can provide an introduction to discrete calculus and demonstrate its effectiveness in many problem domains. This book benefited enormously from the friends and colleagues who provided software, data, and their time in reading the manuscript. In particular, for software we would like to thank Sébastien Bougleux for help with total variation code for an arbitrary graph and Camille Couprie for power watershed code. For use of their data, we wish to thank Jason Bohland, Ja
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